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322       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

are analytic in the interior of D (15.1.5). We shall show that the characters of
A are again in this case the restrictions to A of the Dirac measures sx(x e D).
From this it will follow (15.3.7) that xh^ex is again a homeomorphism of D
onto the spectrum X(A). Let/0 be the canonical injection of D in C ; then it is
enough to show that/0 and the unit element 1 of A generate a dense subalgebra
of A (for it is clear that SpA(/0) = D, and the result will then follow by (1 5.3.6)).
Now, let A0 denote the subalgebra of A consisting of the functions which are
restrictions to D of functions analytic on some neighborhood of D in C. This
subalgebra is dense in A, because for each integer n > 0 and each function
/e A, the function fn which is the restriction to D of the mapping

of ((n + l)/w)D into C clearly belongs to A0, and/rt -/ uniformly on D as
n~+ +00 (3.16.5). On the other hand, every function /eA0 is the uniform
limit (in D) of a sequence of polynomials Pn , namely the partial sums of the
Taylor series for / at the point 0 (9.9.1 and 9.9.2). Hence the set of poly-
nomials in C is dense in A, and our assertion is proved. We remark that here
the Gelfand transformation is isometric, but ^(A) is distinct from

(15.3.9) Let B be the Banach algebra consisting of the restrictions to the
unit circle U : |C| = 1 of the functions belonging to the algebra A of (15.3.8).
Then the restriction mapping/W/| U is an isometric isomorphism of A onto B :
this is an immediate consequence of the maximum modulus principle (9.5.9),
which implies that ||/|| = ||/| U||. Here the mapping xi-+ex of U into X(B) is
again continuous and injective; but it is no longer a homeomorphism, because
all the characters of A can be identified with characters of B, and therefore
X(B) = X(A) = D.

PROBLEMS

1. Let X be a metrizable compact space and B a subalgebra of ^C(X) containing the
unit element. Assume that B is equipped with a norm ||jt ||B which makes it a Banach
algebra.

(a)    Show that ||JC||B ^ W (the norm on ^C<X)) bV remarking that <p(t) :x^x(t)
is a character of B for all / e X. Deduce that 0 is the only quasi-nilpotent element of B.

(b)    Show that 9? is a continuous mapping of X into X(B). If B separates the points
of X (7.3) then 99 is a homeomorphism of X onto a closed subset of X(B).

(c)    Suppose that, for each function x e B, the conjugate x belongs to B, and that if
x E B is such that x(t) ^ 0 for all t e X, then the inverse x"1 of x in ^C(X) belongs to
B. Show that in these conditions the mapping 99 is surjective. (Let m be a maximal
ideal of B, and let Z be the set of all t e X such that x(t)  0 for all x E m. Show that in C, and let A = ^(D) be the